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Generalized Tate Cohomology
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Memoirs of the American Mathematical Society
1995; 178 pp; softcover
Volume: 113
ISBN-10: 0-8218-2603-4
ISBN-13: 978-0-8218-2603-4
List Price: US$50 Individual Members: US$30
Institutional Members: US\$40
Order Code: MEMO/113/543

This book presents a systematic study of a new equivariant cohomology theory $$t(k_G)^*$$ constructed from any given equivariant cohomology theory $$k^*_G$$, where $$G$$ is a compact Lie group. Special cases include Tate-Swan cohomology when $$G$$ is finite and a version of cyclic cohomology when $$G = S^1$$. The groups $$t(k_G)^*(X)$$ are obtained by suitably splicing the $$k$$-homology with the $$k$$-cohomology of the Borel construction $$EG\times _G X$$, where $$k^*$$ is the nonequivariant cohomology theory that underlies $$k^*_G$$. The new theories play a central role in relating equivariant algebraic topology with current areas of interest in nonequivariant algebraic topology. Their study is essential to a full understanding of such "completion theorems" as the Atiyah-Segal completion theorem in $$K$$-theory and the Segal conjecture in cohomotopy. When $$G$$ is finite, the Tate theory associated to equivariant $$K$$-theory is calculated completely, and the Tate theory associated to equivariant cohomotopy is shown to encode a mysterious web of connections between the Tate cohomology of finite groups and the stable homotopy groups of spheres.

Research mathematicians.

• Part II: Eilenberg-Maclane $$G$$-spectra and the spectral sequences
• Appendix A: Splittings of rational $$G$$-spectra for a finite group $$G$$